ARTICLE IN PRESS. Statistics & Probability Letters ( ) A Kolmogorov-type test for monotonicity of regression. Cecile Durot

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1 STAPRO 66 pp: - col.fig.: il ED: MG PROD. TYPE: COM PAGN: Usha.N -- SCAN: il Statistics & Probability Letters Abstract A Kolmogorov-type test for mootoicity of regressio Cecile Durot Laboratoire de Probabilites, Statistique et Modelisatio, Uiversite Paris Sud, Bâtimet 42, 40 Orsay, Cedex, Frace Received August 2002; received i revised form March 200 A ew oparametric procedure for testig mootoicity of a regressio mea is proposed. The test is show to have prescribed asymptotic level ad good asymptotic power. It is based o the supremum distace from a empirical process to its least cocave majorat ad is very easily implemetable. A simulatio study is reported to demostrate ite sample behavior of the procedure. c 200 Published by Elsevier Sciece B.V. MSC: 62G08; 62G0; 62G20 Keywords: Test for mootoicity; Least cocave majorat; Local alterative; Power: Noparametric test. Itroductio We wish to test the ull hypothesis that f is oicreasig agaist the oparametric alterative that it is ot withi the regressio model y i = fx i + i, where x i = i= ad the i s are idepedet ad idetically distributed radom variables with zero mea. The problem of testig mootoicity of a regressio fuctio has bee cosidered i several recet works. Schlee 82 proposed a test based o the greatest discrepacy of a estimator of the derivative of f from zero. Bowma et al. 8 studied a test based o the size of a critical badwidth, the amout of smoothig ecessary to force a oparametric regressio estimator to be mootoe. Hall ad Heckma 2000 poited out that the test of Bowma et al. ca have low power if f possesses a at part, ad they proposed a test based o the slopes of the tted least-squares liear regressio lies over small blocks of observatios. Gijbels et al cosidered a testig procedure which is based o the sigs of y i+k y i. The test statistic proposed by Ghosal et al is a locally weighted versio of Kedall s tau. Baraud et al. 200 cosidered two testig Tel.: ; fax: address: cecile.durot@math.u-psud.fr C. Durot. 06-2/0/$ - see frot matter c 200 Published by Elsevier Sciece B.V. doi:0.06/s

2 STAPRO 66 2 C. Durot / Statistics & ProbabilityLetters procedures. The rst oe is aki to that of Hall ad Heckma, ad is based o local gradiets while the secod oe is based o local meas. Fially, Dumbge ad Spokoiy 200 test mootoicity withi a white oise model by cosiderig the supremum, over all badwidths, of the supremum distace betwee a kerel estimator ad the ull hypothesis. The commo drawback of the above-metioed tests relies o their practical implemetatio. Schlee did ot discuss practical implemetatio, the test proposed by Dumbge ad Spokoii requires the computatio of kerel estimators for all badwidths, ad the other tests ivolve a parameter that is chose i a somewhat arbitrary way. Our aim is to propose a ew test that is easily implemetable ad does ot require ay choice of smoothig parameters. The proposed test is based o the fact that ˆF F if ad oly if f is oicreasig, where Ft= t 0 fsds, t [0; ], ad ˆF is the least cocave majorat of F. It cosists of rejectig the ull hypothesis if the supremum distace betwee a cosistet estimator F of F ad its least cocave majorat ˆF is too large. The test is calibrated o costat fuctios. The paper is orgaised as follows. I Sectio 2, the testig procedure is described ad its asymptotic level ad power are studied. Moreover, a Bootstrap procedure is proposed for cases whe is small. Simulatios are reported i Sectio ad Sectio 4 is devoted to the proofs. 2. The testig procedure We wish to test the ull hypothesis H 0 : f is oicreasig o [0; ] agaist the oparametric alterative that it is ot, o the basis of the observatios y i = fx i + i ; i=;:::;. Here, x i = i=, the i s are i.i.d. radom variables with mea zero ad ite variace 2 0, ad f is a ukow cotiuous fuctio o [0; ]. We assume furthermore E i p for some p Costructio of the test It is assumed that f is cotiuous so f is oicreasig o [0; ] if ad oly if F is cocave o [0; ], where F = 0 fsds. For all t [0; ], let i t deote the iteger part of t ad dee F t= y j +t x it y it+; t [0; ]: It ca be show that F coverges i probability to F i the supremum distace sese so the distace of F to its least cocave majorat l.c.m. ˆF is certaily small if f is oicreasig ad large otherwise. We thus cosider the Kolmogorov-type test statistic S deed by S = sup ˆF t F t: ˆ t [0;] Here, ˆ 2 is a cosistet estimator for 2. Oe may retai for istace ˆ 2 = y i y i+ 2 ; 2 i= 2

3 Table Simulated -quatile q of Z STAPRO 66 C. Durot / Statistics & ProbabilityLetters q see Rice 84. Our test cosists of rejectig H 0 if S exceeds a critical value that we derive from the followig results. I the sequel, deotes supremum distace o [0; ]; W is a stadard Browia motio ad Ŵ is the l.c.m. of {W t;t [0; ]}. Moreover, we set Z = Ŵ W. Theorem. i Costat fuctios are asymptotically least favorable. ii If f is costat the S coverges i distributio to Z as goes to iity. Lemma. The distributio of Z is cotiuous. We thus reject H 0 if S q, where is a xed umber i 0; ad q satises PZ q =. It follows from the previous results that our test has asymptotic level. The test is very easily implemetable: it does ot require ay arbitrary choice of parameters ad oe ca d i the literature algorithms for computig the l.c.m. of a give fuctio, see e.g. the Pool-Adjacet-Violators algorithm PAVA i Barlow et al. 2. Moreover, the quatile q ca be easily estimated via simulatios. Oe ca ideed costruct a Browia motio W over a grid of poits, liearly iterpolate ad costruct Ŵ. Oe ca alteratively compute rst Ŵ by usig Carola ad Dykstra 2002 method ad the costruct a liearly iterpolated versio of the Browia path. Empirical quatiles q computed from a grid of 000 poits, the PAVA ad 0; 000 copies of the supremum distace betwee the iterpolated Browia motio ad its l.c.m. are reported i Table. Note that the proposed test is similar i spirit to the DIP test of Hartiga ad Hartiga 8 for uimodality of a desity fuctio. Our test ad the DIP test share the property that the test statistic is asymptotically positive for costat fuctios ad asymptotically zero for smooth, strictly decreasig away from the mode for the DIP test fuctios, see Theorem. of Durot ad Tocquet 2000 ad Theorems ad of Hartiga ad Hartiga 8. Note moreover that we have cosidered for the sake of simplicity a model with a sigle observatio per desig poit, but that the proposed testig procedure also applies i more geeral models. Assume for istace we observe y ij ; i=;:::;; j=;:::;r, ad the observatios obey the model y ij =fi=+ ij. Here, the ij s are i.i.d. cetered variables with E ij p for some p 2; r is a sequece of positive itegers that satises lim if r 0 ad lim r = = 0, ad f is a ukow Holderia fuctio: fx fy 6 L x y s for all x ad y i [0; ], some L 0 ad some s 0 that satises lim r 2s =0.Ifwesety i = j y ij=r ad dee S by where ˆ 2 is give by 2, the the test that rejects the hypothesis that f is oicreasig if S q has asymptotic level Asymptotic power We study here the asymptotic power of our test agaist a sequece of local alteratives. The regressio mea may deped o so it is deoted here by f istead of f. Moreover,

4 STAPRO 66 4 C. Durot / Statistics & ProbabilityLetters we set E t=e f [F t] = f x j +t x it f x it+; t [0; ]: Theorem 2. For ay 0; there is some 0 such that lim if P f S q ; provided lim if Ê E. Here, Ê deotes the l.c.m. of E. To illustrate the above theorem, let dee C t= t 0 f sds; t [0; ] ad let Ĉ be the l.c.m. of C. We have Ĉ C if ad oly if f is oicreasig, ad Ĉ C measures departure from mootoicity. We thus describe below the asymptotic power of our test i terms of the discrepacy of C to Ĉ.Iff does ot deped o ad is ot oicreasig, the Ĉ C does ot deped o ad is strictly positive. Moreover, f is cotiuous so E C coverges to zero as, ad also, Ê Ĉ coverges to zero as recall that the supremum distace betwee least cocave majorats of fuctios is less tha or equal to the supremum distace betwee the fuctios themselves. I that case, holds for ay 0;, which meas that our test has asymptotic power agaist ay xed, cotiuous alterative. O the other had, if f depeds o ad belogs to aholder smoothess class: f {g; gx gy 6 L x y s for all x; y [0; ]} for some L 0 ad s =2, the C E =O =2. I that case, holds provided lim if Ĉ C for some large eough. 2.. The case of small sample size The asymptotic distributio of S uder the least favorable hypothesis is cotiuous so i the case of small sample size, we recommed to evaluate the critical value of the test via Bootstrap methods. Let i be a Bootstrap versio of i =, draw as follows. I the case of a semiparametric model where i = are i.i.d. variables with kow distributio G, draw ;:::; as a sample from G, idepedet of y ;:::;y. I the case of oparametric models where both f ad G are ukow, cosider a estimator fˆ for f for which i= fˆ x i fx i p = coverges i probability to zero for some p 2. Coditio o y ;:::;y, draw a sample ;:::; from the distributio that puts mass = at each poit y i fˆ x i y j fˆ x j ; 6 i 6 ; j= compute the coditioal stadard deviatio ˆ of i G t= j +t x it i t+; t [0; ]: ad put i = i = ˆ. Dee the 2 2 Let Ĝ be the l.c.m. of G ad S = Ĝ G. Oe ca prove that S coverges i distributio to Z. Ifq;;B deotes the empirical -quatile of S computed from B copies of S, it thus

5 STAPRO 66 C. Durot / Statistics & ProbabilityLetters follows from Lemma that uder H 0 ; lim lim P fs q ;;B 6 B with equality if f is costat. The test that rejects H 0 if S q;;b thus has asymptotic level, ad it is easily see that its asymptotic power as B ad go to iity is greater tha a prescribed 0; uder the assumptios of Theorem 2. Moreover, the distributio of S is close to the distributio of S i the least favorable case, whe f is costat, so we guess that the level of this test is close to eve for quite moderate provided B is large eough, see Sectio for simulatios.. Simulatios We carried out a simulatio study to demostrate ite sample behavior of our tests withi the regressio model y i = fi=+ i ; 6 i 6. We xed =0:0; = 00 ad we cosidered i.i.d. radom variables ;:::; with mea zero, variace ad kow distributio G. Three dieret distributios were cosidered: G N deotes the stadard Gaussia distributio, G T deotes the distributio of T= where T has Studet distributio with degrees of freedom, ad G C deotes the distributio of C = 2, where C has 2 distributio with degree of freedom. Note that is the lowest idex so that E T p is ite for some p 2 ad that G C is asymmetric. We used the PAVA to compute l.c.m. ad cosidered Rice s estimator of 2, see 2. For each G we computed q;;b G, the empirical quatile of B=0; 000 copies of S. We the geerated s = 000 idepedet copies of S for give f ad 0, ad we couted the proportio of rejectios of H 0 for the test deoted by a.l.c.m. ad deed by the critical regio {S q } ad also for the test deoted by l.c.m. ad deed by the critical regio {S q;;b G }. We rst studied the level of our tests by settig ft=at for a give opositive a. Proportios of 2 rejectios of H 0 are give i Table 2. As expected, the proportio of rejectios for l.c.m. is close to the target level =0:0 for all three error distributios whe a=0 which is the least favorable case Table 2 Estimated levels of l.c.m. ad a.l.c.m. a G Critical value Proportio of rejectios l.c.m. a.l.c.m. l.c.m. a.l.c.m G N G T G C G N G T G C :0 G N G T G C

6 STAPRO 66 6 C. Durot / Statistics & ProbabilityLetters Table Regressio fuctios cosidered i the simulatio study f a ft f ;a 0.; 0. at f 2;a 0.0; 0.2 a exp 0t 0: 2 f 0:2t + f 2;0:2t f 4 t t f ;a 0; 0. t 0: I t60: at 0: + exp 20t 0:2 2 { 0t 0: + exp 00t 0:2 2 if t 0: f 6 0:t 0: + exp 00t 0:2 2 otherwise Table 4 Estimated power of tests for mootoicity f Proportio of rejectios for l.c.m. T LM T LG T ru T st T lk G N G T G C G N G N G N G N G N 0.02 f ;0: f ;0: f 2;0: f 2;0: :004 f f f ; f ;0: f for mootoicity. The test thus seems to be robust agaist heavy-tailed ad asymmetric departures from ormality. The estimated level of a.l.c.m. is less tha =0:0, which meas that this test is coservative, but it is far from the target level. L.c.m. is thus better here. Whe a = 0:0, the regressio fuctio is strogly decreasig which leads to small proportio of rejectios for both tests. We the compared the power of l.c.m. to that of other tests for mootoicity. I the sequel, we deote by T ru the test of Gijbels et al based o the rus of equal sigs of diereces; we deote by T LM ad T LG the tests of Baraud et al. 200 based o local meas ad local gradiets; we deote by T st the test of Hall ad Heckma 2000 based o a Studetised statistic, ad by T lk the test of Ghosal et al The regressio fuctios we cosidered are give i Table. I some cases, the cosidered regressio fuctio depeds o a parameter a which is also give i Table. Proportio of rejectios of H 0 obtaied for l.c.m. is give i Table 4 for all three error distributios. Whe give i the literature, proportio of rejectios for the above-metioed tests i a Gaussia model with = 00 are also remided i Table 4. Whatever the error distributio, l.c.m. possesses good power for all studied fuctios f. Its power is comparable to that of other tests, except i the case of f ;a where the power is lower tha that of T LM ; T LG or T st.

7 STAPRO 66 C. Durot / Statistics & ProbabilityLetters 4. Proofs We refer to Sectio 2 i Durot ad Tocquet 2000 for properties of least cocave majorats. Proof of Theorems ad 2. Let G be the process give by G = F E f [F ], that is G t= j +t x it it+; t [0; ] ad let Ĝ be the l.c.m. of G. The fuctio t E f [F t] + Ĝ t is above F ad cocave uder H 0. It is thus also above ˆF so uder H 0 ; S 6 Ĝ G = ˆ.Iff c for some costat c, the F t=g t+ct, which implies that ˆF t=ĝ t+ct. We thus have S = Ĝ G = ˆ if f is costat, which proves that costat fuctios are least favorable. We assume without loss of geerality that the i s are deed o some rich eough probability space so that Sakhaeko s costructio holds, see Sakhaeko 8. I that case, there exists some Browia motio W with sup t [0;] j W t =O P =p =2 =o P ; recall p 2. We have max 6k6 k =o P =2 so G W =o P : 4 The supremum distace betwee least cocave majorats of two processes is less tha or equal to the supremum distace betwee the processes themselves so Ĝ G coverges i distributio to Z as goes to iity. It is assumed that ˆ 2 coverges i probability to 0as so S coverges i distributio to Z whe f is costat, which completes the proof of Theorem. It follows from 4 that F E =O Pf =2, ad ˆ 2 coverges i probability to 0so S = Ê E O Pf ; ˆ which proves Theorem 2. Proof of Lemma. The process {Ŵ t W t; t [0; ]} has cotiuous paths so its supremum is achieved ad we ca dee { } z 0 =sup We dee moreover t [0; ]; Ŵ t W t = sup u 0 =sup { t [0; ; Ŵ t=w t } u [0;] Ŵ u W u : 2 2 ote that Ŵ 0=W 0 so the latter set is ot empty. For every 0;, we ca have u 0 6 oly if there exists some v [0;] such that the lie segmet joiig v; W v ad ;W is above

8 STAPRO 66 8 C. Durot / Statistics & ProbabilityLetters the poit t; W t for every t [0; ]. Moreover, sup v [0;] W v =O P as 0so Pu P sup W t tw 6 O P : t [0;] Restrictig the supremum o [0; ] ad usig scalig property of Browia motio the yields Pu P sup =4 W t 6 O P t [0;] ; which teds to zero as 0 sice sup t [0;] W t is positive ad has bouded desity. It follows that PZ x; z 0 u 0 =PZ x; z 0 u 0 +o ; where o deotes a real umber that teds to zero as 0. For every positive, let Ŵ be the l.c.m. of {W t; t [; ]}; z be the supremum of those poits where sup t [;] Ŵ t W t is achieved, ad u be the supremum of those poits i [; where Ŵ hits W. I the case whe z 0 u 0 we have Ŵ t=ŵ t for all t [u 0 ; ]; z =z 0 ; u =u 0 ad Z =sup t [;] Ŵ t W t. Hece, PZ x; z 0 u 0 6 P sup Ŵ t W t x; z u +o : t [;] If u 0 = the there exists some 0 with Ŵ t=w t for all t [ ; ], so Pu 0 = = 0 ad PZ x; z 0 u 0 =PZ x; z 0 u 0 +o : Let Ŵ 0 be the least cocave majorat of {W t; t [; +]}; y 0 be the supremum of those poits i [; +] where the supremum of {Ŵ 0 t W t; t [; +]} is achieved ad v 0 the supremum of those poits i [; + where Ŵ 0 hits W. By time homogeeity of Browia motio, PZ x; z 0 u 0 =P sup Ŵ 0 t W t x; y 0 v 0 +o : t [;+] If y 0 v 0 the u v 0 ; z = y 0 ad PZ x; z 0 u 0 6 P sup Ŵ t W t x; z u +o : t [;] The latter iequality combied with yields PZ x 6 P sup Ŵ t W t x +o : t [;]

9 STAPRO 66 C. Durot / Statistics & ProbabilityLetters But Ŵ t = is the least cocave majorat at time t of {W = ; [; ]} so scalig property of Browia motio yields P sup Ŵ t W t x = P Z x: t [;] Combiig the two last displays proves that x PZ x is right cotiuous o 0;. This fuctio is also left cotiuous ad PZ = 0 = 0 sice it is the probability for W to be cocave o [0; ]. Therefore, PZ = x = 0 for every x 0, which completes the proof of Lemma. Refereces Baraud, Y., Huet, S., Lauret, B., 200. Tests for covex hypotheses. Preprit, Uiversite Paris Sud, Orsay, Frace. Barlow, R.E., Bartholomew, D.J., Bremer, J.M., Bruk, H.D., 2. Statistical Iferece uder Order Restrictios. Wiley, New York. Bowma, A.W., Joes, M.C., Gijbels, I., 8. Testig mootoicity of regressio. J. Comput. Graph. Statist., Carola, C., Dykstra, R., Characterizatio of the least cocave majorat of Browia motio, coditioal o a vertex poit, with applicatio to costructio. Techical Report. Dumbge, L., Spokoiy, V.G., 200. Multiscale testig of qualitative hypotheses. A. Statist. 2, Durot, C., Tocquet, A.-S., O the distace betwee the empirical process ad its cocave majorat i a mootoe regressio framework. A. I.H.P., to appear. Ghosal, S., Se, A., va der Vaart, A.W., Testig mootoicity of regressio. A. Statist. 28, Gijbels, I., Hall, P., Joes, M.C., Koch, I., Tests for mootoicity of a regressio mea with guarateed level. Biometrika 8, Hall, P., Heckma, N.E., Testig for mootoicity of a regressio mea by calibratig for liear fuctios. A. Statist. 28, 20. Hartiga, J.A., Hartiga, P.M., 8. The dip test of uimodality. A. Statist., Rice, J., 84. Badwidth choice for oparametric kerel regressio. A. Statist. 2, Sakhaeko, A.I., 8. Estimates i the ivariace priciple. Predel ye Teoremy Teorii Veroyatostej. Tr. Ist. Mat., Schlee, W., 82. Noparametric tests of the mootoy ad covexity of regressio. I: Gedeko, B.V., Puri, M.L., Vicze, I. Eds., Noparametric Statistical Iferece, Vol. 2. North-Hollad, Amsterdam, pp